Scattering diagrams, tight gradings, and generalized positivity.
basic_science · Level V
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- Record sourced from PubMed, PMID 40314986.
- Also identified by DOI 10.1073/pnas.2422893122 and PMC identifier 12067233.
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Abstract
In 2013, Lee, Li, and Zelevinsky introduced combinatorial objects called compatible pairs to construct the greedy bases for rank-2 cluster algebras, consisting of indecomposable positive elements including the cluster monomials. Subsequently, Rupel extended this construction to the setting of generalized rank-2 cluster algebras by defining compatible gradings. We find a class of combinatorial objects which we call tight gradings. Using this, we give a directly computable, manifestly positive, and elementary but highly nontrivial formula describing rank-2 consistent scattering diagrams. This allows us to show that the coefficients of the wall-functions on a generalized cluster scattering diagram of any rank are positive, which implies the Laurent positivity for generalized cluster algebras and the strong positivity of their theta bases.